Parameterization of non-linear manifolds

TL;DR

This paper investigates methods for parameterizing low-dimensional manifolds in high-dimensional spaces, focusing on ensuring the parameterization is one-to-one and non-singular to facilitate analysis and applications.

Contribution

It introduces a framework for parameterizing manifolds that guarantees invertibility and non-singularity of the mapping, improving upon existing approaches.

Findings

Proposes a new parameterization method for manifolds.

Ensures the Jacobian of the parameterization is bounded and non-singular.

Facilitates more reliable manifold analysis in high-dimensional data.

Abstract

In this report we consider the parameterization of low-dimensional manifolds that are specified (approximately) by a set of points very close to the manifold in the original high-dimensional space. Our objective is to obtain a parameterization that is (1-1) and non singular (in the sense that the Jacobian of the map between the manifold and the parameter space is bounded and non singular).